Resumen de Weak moduly of convexity

Antonio Ullán de Celis, Javier Alonso Romero

• Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] dE(e) = inf {1 - 1/2||x + y||: x,y Î B, ||x - y|| = e} (0 = e = 2) is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, as dE(e,g) = inf {1 - 1/2||x + y||: x,y Î B, g(x-y) = e} dE(e,f) = inf {1 - 1/2 f(x,y): x,y Î B, ||x - y|| = e} dE(e,f,g) = inf {1 - 1/2 f(x,y): x,y Î B, g(x-y) = e} where 0 = e = 2 and f,g Î S' (unit sphere of the topological dual space E').

  D. F. Cudia [3] has shown the close connection existing between these moduli and various differentiability conditions of the norm in E'.

  In this note we study these moduli from a different point of view, then we analyze some of its properties and we see that it is possible to characterize inner product spaces by means of them.